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## High school geometry

### Course: High school geometry>Unit 9

Lesson 3: Volume and surface area

# Volume formulas review

Review the formulas for the volume of prisms, cylinders, pyramids, cones, and spheres.
It may seem at first like there are lots of volume formulas, but many of the formulas share a common structure.

## Prisms and prism-like figures

start text, V, o, l, u, m, e, end text, start subscript, start text, p, r, i, s, m, end text, end subscript, equals, left parenthesis, start color #0c7f99, start text, b, a, s, e, space, a, r, e, a, end text, end color #0c7f99, right parenthesis, dot, left parenthesis, start color #ca337c, start text, h, e, i, g, h, t, end text, end color #ca337c, right parenthesis
We always measure the height of a prism perpendicularly to the plane of its base. That's true even when a prism is on it's side or when it tilts (an oblique prism).

### Rectangular prisms

Often, we first learn about volume using rectangular prisms (specifically right rectangular prisms), such as by building the prism out of cubes.
Note that any face of a rectangular prism could be its base, as long as we measure the height of the prism perpendicularly to that face.
A cube with a length of l, a width of w, and a height of h.
\begin{aligned} \text{Volume}_{\text{rectangular prism}}&=(\blueE{\text{Area}_{\text{rectangle}}})\cdot (\maroonD{\text{height}})\\\\ &=\left(\blueE{(\text{rectangle base})(\text{rectangle height})}\right)\cdot (\maroonD{\text{prism height}})\\\\ &=\blueE{lw}\maroonD{h} \end{aligned}

### Triangular prisms

A triangular prism has a base shaped like a triangle.
A triangular prism with a triangular base of base b, a triangular base of height h, and the length of l.
\begin{aligned} \text{Volume}_{\text{triangular prism}}&=(\blueE{\text{Area}_{\text{triangle}}})\cdot (\maroonD{\text{height}})\\\\ &=\left(\blueE{\dfrac{1}{2}(\text{triangle base})(\text{triangle height})}\right)\cdot (\maroonD{\text{prism height}})\\\\ &=\blueE{\dfrac{1}{2}bh}\maroonD{\ell} \end{aligned}

### Cylinders

A circular cylinder is a prism-like figure that has a base shaped like a circle.
A cylinder with a radius r and a height h.
\begin{aligned} \text{Volume}_{\text{circular cylinder}}&=(\blueE{\text{Area}_{\text{circle}}})\cdot (\maroonD{\text{height}})\\\\ &=(\blueE{\pi \cdot (\text{radius})^2})\cdot (\maroonD{\text{height}})\\\\ &=\blueE{\pi r^2}\maroonD{h} \end{aligned}

### Oblique prisms

In oblique prisms, the bases are in parallel planes,
We still calculate the volume in exactly the same way because of Cavalieri's principle.
Which expression gives the volume of the oblique rectangular prism?
A oblique rectangular prism with its rectangular base with a length of two units and a width of one point five units. The prism's slanted height is five units. Its vertical height is four units.

## Pyramids and pyramid-like figures

start text, V, o, l, u, m, e, end text, start subscript, start text, p, y, r, a, m, i, d, end text, end subscript, equals, start color #7854ab, start fraction, 1, divided by, 3, end fraction, end color #7854ab, left parenthesis, start color #0c7f99, start text, b, a, s, e, space, a, r, e, a, end text, end color #0c7f99, right parenthesis, dot, left parenthesis, start color #ca337c, start text, h, e, i, g, h, t, end text, end color #ca337c, right parenthesis
We also measure the height of a pyramid perpendicularly to the plane of its base. Because of Cavalieri's principle, the same volume formula works for right and oblique pyramid-like figures.

### Rectangle-based pyramids

A rectangle-based pyramid has a base shaped like a rectangle.
A rectangular pyramid with the rectangular base's length of l units and width of w units. It has a vertical height of h units.
\begin{aligned} \text{Volume}_{\text{rectangle-based pyramid}}&=\purpleD{\dfrac{1}{3}}(\blueE{\text{Area}_{\text{rectangle}}})\cdot (\maroonD{\text{height}})\\\\ &=\purpleD{\dfrac{1}{3}}\left(\blueE{(\text{rectangle base})(\text{rectangle height})}\right)\cdot (\maroonD{\text{pyramid height}})\\\\ &=\purpleD{\dfrac{1}{3}}\blueE{lw}\maroonD{h} \end{aligned}

### Cones

A circular cone is a pyramid-like figure that has a base shaped like a circle.
A cone with a radius of r and a vertical height of h.
\begin{aligned} \text{Volume}_{\text{circular cone}}&=\purpleD{\dfrac{1}{3}}(\blueE{\text{Area}_{\text{circle}}})\cdot (\maroonD{\text{height}})\\\\ &=\purpleD{\dfrac{1}{3}}(\blueE{\pi \cdot (\text{radius})^2})\cdot (\maroonD{\text{height}})\\\\ &=\purpleD{\dfrac{1}{3}}\blueE{\pi r^2}\maroonD{h} \end{aligned}

### Spheres

A sphere with a radius of r.
start text, V, o, l, u, m, e, end text, start subscript, start text, s, p, h, e, r, e, end text, end subscript, equals, start color #a75a05, start fraction, 4, divided by, 3, end fraction, end color #a75a05, pi, left parenthesis, start color #0c7f99, start text, r, a, d, i, u, s, end text, end color #0c7f99, right parenthesis, cubed

## Want to join the conversation?

• Why is the Volume of sphere is divided by 3?
• This is a really good video that I found:

It explains the formula with simple geometry and algebra, and in animation.
• if a square is writen X^2 how is a cube writen
• A number cubed, or in this case x, is written as x³, or x^3.
• When using the pi symbol in calculations, the answer comes out differently than if you just use 3.14, which is how the answers on here are calculated. Why not just recommended to use only 3.14 for pi, not the symbol on calculators as shown on the videos? It is frustrating to get sent back to zero when you use your calculator correctly but you should just use one standard calculation of 3.14.
• how can I find volume of a tetrahedron
• A tetrahedron is just a triangular pyramid. If any pyramid has height h and a base of area b, then it's volume is bh/3.
• What would be the equation to find the volume for a trapezoid prism?
• Any prism volume is V = BH where B is area of base and H is height of prism, so find area of the base by B = 1/2 h(b1+b2), then multiply by the height of the prism.
• What would be the difference between a triangular prism and a pyramid?
• A prism has the base that goes through the whole figure, so a triangular prism will have two congruent bases that are a triangle. A triangular pyramid has a triangular base and each base vertex that has an edge that goes to a common vertex.
• This article shows how to find the volume of a four sided pyramid, but how do you find the volume of a three sided pyramid.
Thanks.